The arithmetic mean — the geometric mean and related matrix inequalities

The difficulty of establishing a (noncommutative) matrix inequality involving the geometric mean was discussed in 1978 by K.V. Bhagwat and R. Subramanian [9] who pointed out that the problem of defining a geometric mean for non-commutative operators “makes it difficult to establish the validity or otherwise of the classical inequalities involving the geometric mean”. However, in a recent paper, Sagae and Tanabe [32] define a geometric mean and establish an AG-GM inequality for a finite number of positive definite matrices. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.